# An Operational Environment for Quantum Self-Testing

Matthias Christandl, Nicholas Gauguin Houghton-Larsen, and Laura Mancinska

Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2200 Copenhagen, Denmark

### Abstract

Observed quantum correlations are known to determine in certain cases the underlying quantum state and measurements. This phenomenon is known as $\textit{(quantum) self-testing}$.
Self-testing constitutes a significant research area with practical and theoretical ramifications for quantum information theory. But since its conception two decades ago by Mayers and Yao, the common way to rigorously formulate self-testing has been in terms of operator-algebraic identities, and this formulation lacks an operational interpretation. In particular, it is unclear how to formulate self-testing in other physical theories, in formulations of quantum theory not referring to operator-algebra, or in scenarios causally different from the standard one.
In this paper, we explain how to understand quantum self-testing operationally, in terms of causally structured dilations of the input-output channel encoding the correlations. These dilations model side-information which leaks to an environment according to a specific schedule, and we show how self-testing concerns the relative strength between such scheduled leaks of information. As such, the title of our paper has double meaning: we recast conventional quantum self-testing in terms of information-leaks to an environment – and this realises quantum self-testing as a special case within the surroundings of a general operational framework.
Our new approach to quantum self-testing not only supplies an operational understanding apt for various generalisations, but also resolves some unexplained aspects of the existing definition, naturally suggests a distance measure suitable for robust self-testing, and points towards self-testing as a modular concept in a larger, cryptographic perspective.

### ► References

[1] Samson Abramsky and Bob Coecke, A Categorical Semantics of Quantum Protocols, Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004, IEEE, 2004, https:/​/​doi.org/​10.1109/​LICS.2004.1319636.
https:/​/​doi.org/​10.1109/​LICS.2004.1319636

[2] John C. Baez, Quantum Quandaries: A Category-Theoretic Perspective, The structural foundations of quantum gravity (2006), 240–265, https:/​/​doi.org/​10.1093/​acprof:oso/​9780199269693.003.0008.
https:/​/​doi.org/​10.1093/​acprof:oso/​9780199269693.003.0008

[3] Jonathan Barrett, Information Processing in Generalized Probabilistic Theories, Physical Review A 75 (2007), no. 3, 032304, https:/​/​doi.org/​10.1103/​PhysRevA.75.032304.
https:/​/​doi.org/​10.1103/​PhysRevA.75.032304

[4] John S. Bell, On the Einstein Podolsky Rosen paradox, Physics Physique Fizika 1 (1964), no. 3, 195–200, https:/​/​doi.org/​10.1103/​PhysicsPhysiqueFizika.1.195.
https:/​/​doi.org/​10.1103/​PhysicsPhysiqueFizika.1.195

[5] Cyril Branciard, Nicolas Gisin, and Stefano Pironio, Characterizing the nonlocal correlations created via entanglement swapping, Physical review letters 104 (2010), no. 17, 170401, https:/​/​doi.org/​10.1103/​PhysRevLett.104.170401.
https:/​/​doi.org/​10.1103/​PhysRevLett.104.170401

[6] Cédric Bamps, Serge Massar, and Stefano Pironio, Device-independent randomness generation with sublinear shared quantum resources, Quantum 2 (2018), 86, https:/​/​doi.org/​10.22331/​q-2018-08-22-86.
https:/​/​doi.org/​10.22331/​q-2018-08-22-86

[7] Cyril Branciard, Denis Rosset, Nicolas Gisin, and Stefano Pironio, Bilocal versus nonbilocal correlations in entanglement-swapping experiments, Physical Review A 85 (2012), no. 3, 032119, https:/​/​doi.org/​10.1103/​PhysRevA.85.032119.
https:/​/​doi.org/​10.1103/​PhysRevA.85.032119

[8] Howard Barnum and Alexander Wilce, Information Processing in Convex Operational Theories, Electronic Notes in Theoretical Computer Science 270 (2011), no. 1, 3–15, https:/​/​doi.org/​10.1016/​j.entcs.2011.01.002.
https:/​/​doi.org/​10.1016/​j.entcs.2011.01.002

[9] Howard Barnum and Alexander Wilce, Post-Classical Probability Theory, Quantum Theory: Informational Foundations and Foils, Springer, 2016, https:/​/​doi.org/​10.1007/​978-94-017-7303-4_11.
https:/​/​doi.org/​10.1007/​978-94-017-7303-4_11

[10] Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti, Theoretical Framework for Quantum Networks, Physical Review A 80 (2009), no. 2, 022339, https:/​/​doi.org/​10.1103/​PhysRevA.80.022339.
https:/​/​doi.org/​10.1103/​PhysRevA.80.022339

[11] Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti, Probabilistic Theories with Purification, Physical Review A 81 (2010), no. 6, 062348, https:/​/​doi.org/​10.1103/​PhysRevA.81.062348.
https:/​/​doi.org/​10.1103/​PhysRevA.81.062348

[12] Giulio Chiribella, Giacomo Mauro D’Ariano, and Paolo Perinotti, Informational derivation of quantum theory, Physical Review A 84 (2011), no. 1, 012311, https:/​/​doi.org/​10.1103/​PhysRevA.84.012311.
https:/​/​doi.org/​10.1103/​PhysRevA.84.012311

[13] Matthias Christandl, Roberto Ferrara, and Karol Horodecki, Upper bounds on device-independent quantum key distribution, Physical Review Letters 126 (2021), no. 16, 160501, https:/​/​doi.org/​10.1103/​PhysRevLett.126.160501.
https:/​/​doi.org/​10.1103/​PhysRevLett.126.160501

[14] Bob Coecke, Tobias Fritz, and Robert W. Spekkens, A mathematical theory of resources, Information and Computation 250 (2016), 59–86, https:/​/​doi.org/​10.1016/​j.ic.2016.02.008.
https:/​/​doi.org/​10.1016/​j.ic.2016.02.008

[15] Andrea Coladangelo, Alex B Grilo, Stacey Jeffery, and Thomas Vidick, Verifier-on-a-leash: new schemes for verifiable delegated quantum computation, with quasilinear resources, Annual International Conference on the Theory and Applications of Cryptographic Techniques, Springer, 2019, https:/​/​doi.org/​10.1007/​978-3-030-17659-4_9.
https:/​/​doi.org/​10.1007/​978-3-030-17659-4_9

[16] Giulio Chiribella, Distinguishability and copiability of programs in general process theories, Int J Software Informatics 1 (2014), no. 2, 209–223, https:/​/​doi.org/​10.48550/​arXiv.1411.3035.
https:/​/​doi.org/​10.48550/​arXiv.1411.3035

[17] John F. Clauser, Michael A. Horne, Abner Shimony, and Richard A. Holt, Proposed experiment to test local hidden-variable theories, Physical review letters 23 (1969), no. 15, 880–884, https:/​/​doi.org/​10.1103/​PhysRevLett.23.880.
https:/​/​doi.org/​10.1103/​PhysRevLett.23.880

[18] Boris S. Cirel'son, Some results and problems on quantum Bell-type inequalities, Hadronic Journal Supplement 8 (1993), no. 4, 329–345.

[19] Roger Colbeck and Adrian Kent, Private randomness expansion with untrusted devices, Journal of Physics A: Mathematical and Theoretical 44 (2011), no. 9, 095305, https:/​/​doi.org/​10.1088/​1751-8113/​44/​9/​095305.
https:/​/​doi.org/​10.1088/​1751-8113/​44/​9/​095305

[20] Bob Coecke and Raymond Lal, Causal Categories: Relativistically interacting Processes, Foundations of Physics 43 (2013), no. 4, 458–501, https:/​/​doi.org/​10.1007/​s10701-012-9646-8.
https:/​/​doi.org/​10.1007/​s10701-012-9646-8

[21] Bob Coecke, Terminality implies non-signalling, arXiv preprint (2014), https:/​/​doi.org/​10.48550/​arXiv.1405.3681.
https:/​/​doi.org/​10.48550/​arXiv.1405.3681

[22] Roger Colbeck, Quantum and relativistic protocols for secure multi-party computation, Ph.D. thesis, University of Cambridge, 2006, https:/​/​doi.org/​10.48550/​arXiv.0911.3814.
https:/​/​doi.org/​10.48550/​arXiv.0911.3814

[23] Andrea Coladangelo, A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations, Quantum 4 (2020), 282, https:/​/​doi.org/​10.22331/​q-2020-06-18-282.
https:/​/​doi.org/​10.22331/​q-2020-06-18-282

[24] Andrea Coladangelo and Jalex Stark, Unconditional separation of finite and infinite-dimensional quantum correlations, arXiv preprint (2018), https:/​/​doi.org/​10.48550/​arXiv.1804.05116.
https:/​/​doi.org/​10.48550/​arXiv.1804.05116

[25] Matthew Coudron and Henry Yuen, Infinite randomness expansion with a constant number of devices, Proceedings of the forty-sixth annual ACM symposium on Theory of computing, 2014, https:/​/​doi.org/​10.1145/​2591796.2591873.
https:/​/​doi.org/​10.1145/​2591796.2591873

[26] Igor Devetak and Peter W. Shor, The capacity of a quantum channel for simultaneous transmission of classical and quantum information, Communications in Mathematical Physics 256 (2005), no. 2, 287–303, https:/​/​doi.org/​10.1007/​s00220-005-1317-6.
https:/​/​doi.org/​10.1007/​s00220-005-1317-6

[27] Artur K. Ekert, Quantum Cryptography based on Bell’s Theorem, Physical Review Letters 67 (1991), no. 6, 661–663, https:/​/​doi.org/​10.1103/​PhysRevLett.67.661.
https:/​/​doi.org/​10.1103/​PhysRevLett.67.661

[28] Tobias Fritz, Beyond bell's theorem: correlation scenarios, New Journal of Physics 14 (2012), no. 10, 103001, https:/​/​doi.org/​10.1088/​1367-2630/​14/​10/​103001.
https:/​/​doi.org/​10.1088/​1367-2630/​14/​10/​103001

[29] Tobias Fritz, A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics, Advances in Mathematics 370 (2020), 107239, https:/​/​doi.org/​10.1016/​j.aim.2020.107239.
https:/​/​doi.org/​10.1016/​j.aim.2020.107239

[30] Koon Tong Goh, Jędrzej Kaniewski, Elie Wolfe, Tamás Vértesi, Xingyao Wu, Yu Cai, Yeong-Cherng Liang, and Valerio Scarani, Geometry of the set of quantum correlations, Physical Review A 97 (2018), no. 2, 022104, https:/​/​doi.org/​10.1103/​PhysRevA.97.022104.
https:/​/​doi.org/​10.1103/​PhysRevA.97.022104

[31] Lucien Hardy, Quantum theory from five reasonable axioms, arXiv preprint (2001), https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​0101012.
https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​0101012
arXiv:quant-ph/0101012

[32] Lucien Hardy, A formalism-local framework for general probabilistic theories, including quantum theory, Mathematical Structures in Computer Science 23 (2013), no. 2, 399–440, https:/​/​doi.org/​10.1017/​S0960129512000163.
https:/​/​doi.org/​10.1017/​S0960129512000163

[33] Nicholas Gauguin Houghton-Larsen, A Mathematical Framework for Causally Structured Dilations and its Relation to Quantum Self-Testing, Ph.D. thesis, University of Copenhagen, 2021, https:/​/​doi.org/​10.48550/​arXiv.2103.02302.
https:/​/​doi.org/​10.48550/​arXiv.2103.02302

[34] Karol Horodecki and Gláucia Murta, Bounds on quantum nonlocality via partial transposition, Physical Review A 92 (2015), no. 1, 010301, https:/​/​doi.org/​10.1103/​PhysRevA.92.010301.
https:/​/​doi.org/​10.1103/​PhysRevA.92.010301

[35] Rahul Jain, Carl A. Miller, and Yaoyun Shi, Parallel device-independent quantum key distribution, IEEE transactions on information theory 66 (2020), no. 9, 5567–5584, https:/​/​doi.org/​10.1109/​TIT.2020.2986740.
https:/​/​doi.org/​10.1109/​TIT.2020.2986740

[36] Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen, MIP*=RE, Commun. ACM 64 (2021), no. 11, 131–138, https:/​/​doi.org/​10.1145/​3485628.
https:/​/​doi.org/​10.1145/​3485628

[37] Jędrzej Kaniewski, Self-testing of binary observables based on commutation, Physical Review A 95 (2017), no. 6, 062323, https:/​/​doi.org/​10.1103/​PhysRevA.95.062323.
https:/​/​doi.org/​10.1103/​PhysRevA.95.062323

[38] Dennis Kretschmann, Dirk Schlingemann, and Reinhard F. Werner, A continuity theorem for Stinespring's dilation, Journal of Functional Analysis 255 (2008), no. 8, 1889–1904, https:/​/​doi.org/​10.1016/​j.jfa.2008.07.023.
https:/​/​doi.org/​10.1016/​j.jfa.2008.07.023

[39] Dennis Kretschmann, Dirk Schlingemann, and Reinhard F. Werner, The information-disturbance tradeoff and the continuity of Stinespring's representation, IEEE transactions on information theory 54 (2008), no. 4, 1708–1717, https:/​/​doi.org/​10.1109/​TIT.2008.917696.
https:/​/​doi.org/​10.1109/​TIT.2008.917696

[40] Aleks Kissinger and Sander Uijlen, A Categorical Semantics for Causal Structure, 2017 32nd Annual ACM/​IEEE Symposium on Logic in Computer Science (LICS), IEEE, 2017, https:/​/​doi.org/​10.23638/​LMCS-15(3:15)2019.
https:/​/​doi.org/​10.23638/​LMCS-15(3:15)2019

[41] Matthew McKague, Interactive Proofs for BQP via Self-Tested Graph States, Theory of Computing 12 (2016), no. 3, 1–42, https:/​/​doi.org/​10.4086/​toc.2016.v012a003.
https:/​/​doi.org/​10.4086/​toc.2016.v012a003

[42] Saunders Mac Lane, Categories for the Working Mathematician, vol. 5, Springer Science & Business Media, 2013, https:/​/​doi.org/​10.1007/​978-1-4757-4721-8.
https:/​/​doi.org/​10.1007/​978-1-4757-4721-8

[43] Carl A. Miller and Yaoyun Shi, Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices, Journal of the ACM (JACM) 63 (2016), no. 4, 1–63, https:/​/​doi.org/​10.1145/​2885493.
https:/​/​doi.org/​10.1145/​2885493

[44] Dominic Mayers and Andrew Yao, Quantum Cryptography with Imperfect Apparatus, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No. 98CB36280), IEEE, 1998, https:/​/​doi.org/​10.1109/​sfcs.1998.743501.
https:/​/​doi.org/​10.1109/​sfcs.1998.743501

[45] Dominic Mayers and Andrew Yao, Self testing quantum apparatus, Quantum Info. Comput. 4 (2004), no. 4, 273–286, https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​0307205.
https:/​/​doi.org/​10.48550/​arXiv.quant-ph/​0307205
arXiv:quant-ph/0307205

[46] Matthew McKague, Tzyh Haur Yang, and Valerio Scarani, Robust self-testing of the singlet, Journal of Physics A: Mathematical and Theoretical 45 (2012), no. 45, 455304, https:/​/​doi.org/​10.1088/​1751-8113/​45/​45/​455304.
https:/​/​doi.org/​10.1088/​1751-8113/​45/​45/​455304

[47] Mark Naimark, Spectral functions of a symmetric operator, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya 4 (1940), no. 3, 277–318.

[48] Anand Natarajan and Thomas Vidick, A quantum linearity test for robustly verifying entanglement, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, 2017, https:/​/​doi.org/​10.1145/​3055399.3055468.
https:/​/​doi.org/​10.1145/​3055399.3055468

[49] Anand Natarajan and John Wright, NEEXP is contained in MIP, 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), IEEE, 2019, https:/​/​doi.org/​10.1109/​FOCS.2019.00039.
https:/​/​doi.org/​10.1109/​FOCS.2019.00039

[50] Paolo Perinotti, Causal Structures and the Classification of Higher Order Quantum Computations, Time in physics, Springer, 2017, https:/​/​doi.org/​10.1007/​978-3-319-68655-4_7.
https:/​/​doi.org/​10.1007/​978-3-319-68655-4_7

[51] Marc-Olivier Renou and Salman Beigi, Nonlocality for generic networks, Phys. Rev. Lett. 128 (2022), 060401, https:/​/​doi.org/​10.1103/​PhysRevLett.128.060401.
https:/​/​doi.org/​10.1103/​PhysRevLett.128.060401

[52] Ben W. Reichardt, Falk Unger, and Umesh Vazirani, Classical Command of Quantum Systems, Nature 496 (2013), no. 7446, 456–460, https:/​/​doi.org/​10.1038/​nature12035.
https:/​/​doi.org/​10.1038/​nature12035

[53] Ivan Šupić, Remigiusz Augusiak, Alexia Salavrakos, and Antonio Acín, Self-testing protocols based on the chained Bell inequalities, New Journal of Physics 18 (2016), no. 3, 035013, https:/​/​doi.org/​10.1088/​1367-2630/​18/​3/​035013.
https:/​/​doi.org/​10.1088/​1367-2630/​18/​3/​035013

[54] Ivan Šupić and Joseph Bowles, Self-testing of quantum systems: a review, Quantum 4 (2020), 337, https:/​/​doi.org/​10.22331/​q-2020-09-30-337.
https:/​/​doi.org/​10.22331/​q-2020-09-30-337

[55] Peter Selinger, Towards a Semantics for Higher-Order Quantum Computation, Proceedings of the 2nd International Workshop on Quantum Programming Languages, TUCS General Publication, vol. 33, 2004, pp. 127–143.

[56] David Schmid, Thomas C. Fraser, Ravi Kunjwal, Ana Belen Sainz, Elie Wolfe, and Robert W. Spekkens, Understanding the interplay of entanglement and nonlocality: motivating and developing a new branch of entanglement theory, arXiv preprint (2020), https:/​/​doi.org/​10.48550/​arXiv.2004.09194.
https:/​/​doi.org/​10.48550/​arXiv.2004.09194

[57] William Forrest Stinespring, Positive functions on C*-algebras, Proceedings of the American Mathematical Society 6 (1955), no. 2, 211–216, https:/​/​doi.org/​10.2307/​2032342.
https:/​/​doi.org/​10.2307/​2032342

[58] Marco Tomamichel, Roger Colbeck, and Renato Renner, Duality between smooth min- and max-entropies, IEEE Transactions on information theory 56 (2010), no. 9, 4674–4681, https:/​/​doi.org/​10.1109/​TIT.2010.2054130.
https:/​/​doi.org/​10.1109/​TIT.2010.2054130

[59] Armin Tavakoli, Máté Farkas, Denis Rosset, Jean-Daniel Bancal, and Jedrzej Kaniewski, Mutually unbiased bases and symmetric informationally complete measurements in Bell experiments, Science Advances 7 (2021), no. 7, https:/​/​doi.org/​10.1126/​sciadv.abc3847.